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A180047
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Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/3 + w/...
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4
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0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 6, 0, 120, 36, 1, 0, 720, 240, 12, 0, 5040, 1800, 120, 1, 0, 40320, 15120, 1200, 20, 0, 362880, 141120, 12600, 300, 1, 0, 3628800, 1451520, 141120, 4200, 30, 0, 39916800, 16329600, 1693440, 58800, 630, 1, 0, 479001600
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OFFSET
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0,5
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COMMENTS
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Equivalence to the binomial formula needs formal proof. This c.f. converges to A052119 = 0.697774657964.. = BesselI(1,2)/BesselI(0,2) for w = 1.
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LINKS
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FORMULA
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T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.
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EXAMPLE
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Triangle starts:
0;
0, 1;
0, 2;
0, 6, 1;
0, 24, 6;
0, 120, 36, 1;
0, 720, 240, 12;
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The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.
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MATHEMATICA
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Table[CoefficientList[Numerator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k, {k, 1, n}]]]], w], {n, 16}]; (* or equivalently *) Table[(n-m+1)!/m! *Binomial[n-m, m-1], {n, 0, 16}, {m, 0, Floor[n/2+1/2]}]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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